Determine the remaining Laplacian eigenvalues

Determine the remaining eigenvalues of the quotient matrix associated with the Laplacian matrix of the generating graph of Z_n, beyond the eigenvalues already identified and the bounds obtained through Weyl's inequalities.

Background

The Laplacian spectrum is expressed as a collection of explicitly known eigenvalues together with the spectrum of a smaller symmetric quotient matrix L_Q. The authors identify 0, φ(n), and n as eigenvalues of L_Q and derive upper and lower bounds for its remaining eigenvalues using the matrix decomposition L_Q = -\tilde Q_n + D_L. The complete determination of the remaining eigenvalues is left unresolved because the authors find solving for them challenging.

References

Since it is equivalent to determining the spectrum of the matrices L_Q and M . We still haven't get all the eigenvalues, as we can see the remaining eigenvalues proves to be challenging. Therefore, we aim to establish bounds for the eigenvalues of L_Q .

— Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$  (2501.09771 - Samant et al., 16 Jan 2025) in Section 5, Laplacian Matrix Spectrum

Is it possible to analyze $Round(\chi*)$ directly instead of upper-bounding it by the kernel cut in order to obtain a general factor smaller than $q/(q^-1)$ or a bound that depends on the quotient generator distribution?

— Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs  (2609.05368 - Stamoulis, 4 Sep 2026) in Section 6, “Discussion and open problems,” second Question